For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
votes
2answers
186 views

What is determinant? [duplicate]

I know this can be the most stupid question here. However, what I want to ask is not how to compute determinant or what the definition of determinant is.(Enough homework :P ) What I really want to ...
1
vote
1answer
162 views

homework - Find a basis for the space of all vectors in R6 with x1 + x2 = x3+ x4 = x5+ x6

a) Find a basis for the space of all vectors in $\mathbb{R}^6 $ with $x_1 + x_2 = x_3 + x_4 = x_5 + x_6$. b) Find a matrix with that subspace as null space. c) Find a matrix with that subspace as ...
3
votes
0answers
65 views

The eigenvector of Laplacian matrix plus a rank one matrix

Denote $L=\left[\begin{array}{ccc} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{array}\right]$ and $M=\left[\begin{array}{ccc} 1\\ & 0\\ & & 0 \end{array}\right]$. ...
0
votes
1answer
52 views

Matrix inversion for a $3\times3$ matix

How does one show that a $3\times3$ matrix is invertible? The matrix is: $$ \left(\begin{array}{ccc} \cos X & -\sin X & 0 \\ \sin X & \cos X &0 \\ 0 & 0 & 1 ...
2
votes
0answers
79 views

Determining the matrix of a linear transformation

Let $A$ be an $n\times n$ matrix, and let $V$ denote the space of $n$-dimensional row vectors. What is the matrix of the linear operator ‘‘right multiplication by $A$’’ with respect to the standard ...
25
votes
2answers
1k views

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : ...
1
vote
1answer
259 views

the identity matrix is unique

let A be an $m\times n$ matrix. Prove that there are unique matrices $I_m$ and $I_n$ such that : $$I_mA=AI_n=A$$ Actually I can't prove the uniqueness here,any help is appreciated. Thanks
0
votes
2answers
66 views

Linearly independent matrix proof

Suppose that $S=\{u_1,u_2,…,u_n\}$ is a set of vector from $\mathbb{R}^m$. Show that $S$ is linearly independent if and only if the set $S'=\left\{u_1,\ \sum_{i=1}^2 u_i,\ \sum_{i=1}^3 u_i,\ \ldots,\ ...
1
vote
3answers
55 views

Why the columns of $A$ are linearly dependent set?

If $A_{m\times n}$ is a matrix such that $\sum_{j=1}^n a_{ij}=0$ for each $i=1,2,…,m,$ then why the columns of $A$ are linearly dependent set, and hence $\operatorname {rank}(A)<n$?
1
vote
0answers
47 views

Matrix exponent and representations of $\mathbb{R}$

It is well known that $\exp(C^{-1}AC)=C^{-1}\exp(A)C$, where $C$ is an invertible matrix. Really, $$\exp(A)=\sum{}\frac{A^k}{k!} \quad \text{and} \quad (C^{-1}AC)^k=C^{-1}A^kC,$$ so the first ...
1
vote
0answers
55 views

Given an n x n matrix A, is there an n x n matrix E such that $A\odot E=A$ and $A \odot F=E$?

For two matrices of dimensions $m \times n$ and $n \times k$, define $C=A\odot B$ to be the matrix with entries $$C_{ij}=\max_{k=1}^n A_{ik} + B_{kj}$$. Given an $n \times n$ matrix $A$, is there ...
0
votes
1answer
77 views

about diagonal matrix and eigenvalues

I am reading the introduce of linear system and eigenvalues. There I read if there is a matrix $A$ and vector $x$, it could find a eigenvalue $\lambda$ such that $$Ax = \lambda x$$ I have a really ...
0
votes
1answer
40 views

Matrix column addition

Suppose you have a matrix in the form of: $$\left[\begin{array}{c} a\\ b\end{array}\right]$$ How can this be represented be a two by two matrix?
3
votes
0answers
38 views

The singularity of a family of matrices

Let $l\ge2$ be an even integer, $\zeta$ be a primitive $l$th root of unity in $\mathbb{C}$. Is it true for any $\alpha=(\alpha_1,\dots,\alpha_l)$ and $\beta=(\beta_1,\dots,\beta_l)$ such that ...
1
vote
2answers
72 views

Help with calculating the determinant

Does anyone know how to go about answering the following? Any help is appreciated! Calculate the determinant of $D = \begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$ and use it to find ...
1
vote
1answer
54 views

Does $\mathbf{W}^H\mathbf{W}=\mathbf{I}$ imply $\mathbf{WW}^H=\mathbf{I}$?

Does $\mathbf{W}^H\mathbf{W}=\mathbf{I}$ imply $\mathbf{WW}^H=\mathbf{I}$? Note: $\mathbf{W}$ is a square complex constant matrix, $\mathbf{W}^H$ is the conjugate transpose of $\mathbf{W}$, and ...
1
vote
1answer
896 views

Why do we assume that a matrix in quadratic form is Symmetric?

I am looking to the review document for linear algebra and the part of the quadratic form (pg17) mentions about an assumption of being symmetric for a matrix in quadratic form. It also includes some ...
0
votes
2answers
100 views

Quotient ring of $T/I$

Please help me to identify the Quotient Ring of $T/I$, since $T$ is set of all triangular matrices, and $I$ is set of all strictly triangular matrices and $I$ is ideal in $T$. For your help I am ...
2
votes
1answer
516 views

How to solve the matrix equation $ABA^{-1}=C$ with $\operatorname{Tr}(A)=a$

I have the following matrix equation: $$ABA^{-1}=C$$ with $B$ and $C$ given and $A$ unknown. The constraint on $A$ is $\operatorname{Tr}(A)=a$ with $a\in\mathbb{R}$. The matrices are $N\times N$.
8
votes
1answer
165 views

Expressing the values of a matrix at pow N

I have a square matrix (that comes from a Markov Chain) that looks like that: $$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 ...
1
vote
1answer
120 views

Finding a mapping such that its kernel equals the image of another non bijective mapping

For an $a \in \mathbb{R}$ let $\phi_a: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear mapping such that $\phi_a(x) := \begin{pmatrix} 1 & 2 & 2 \\1 & 3 & 5 \\ 1 & -1 & a ...
2
votes
2answers
126 views

How to solve this system of linear equations

$$M = \left(\begin{smallmatrix} a_1 & a_2 & a_3 & a_4\\ b_1 & b_2 & b_3 & b_4\\ a_1 & c_2 & b_2 & c_4\\ a_4 & d_2 & b_3 & c_4\\ b_1 & c_2 & a_2 ...
1
vote
1answer
49 views

Consequences of the properties of B on the results of a Generalized Eigenvalue $\lambda$ B v=Av

I'm trying to find a good source for the consequences of the properties of the matrix $B$ for the generalized eigenvalue problem: $\lambda B v = A v \Leftrightarrow \textrm{eig}(B^{-1}A)$ For ...
0
votes
4answers
220 views

Can't solve 3 variables - Systems of Linear Equations

I've was asked to solve this (as homework): $$2x + y + z = 3$$ $$4x + 2z = 10$$ $$2x + 2y = -2$$ I need to solve it with matrices and I have NO IDEA how to do so. I need your help. thanks.
3
votes
1answer
97 views

Commutants for collections of elementary matrices

Let ${\mathbb K}={\mathbb R}$ or $\mathbb C$. Let $V$ be a vector space over $\mathbb K$ and fix a basis $\cal B$ of $V$. We say that a family of vectors of $V$ is nice (relatively to $\cal B$) if it ...
4
votes
2answers
205 views

Two matrices of complementary rank that sum to the identity have zero product.

Suppose $A$, $B$ are real $n \times n$ matrices with $A + B = I$ and rank A + rank B = n. How can one show that $AB = BA = 0$?
2
votes
1answer
388 views

Wikipedia Proof of Skolem-Noether Theorem

Could someone help me understand the wikipedia proof of the Skolem-Noether theorem? In particular, I'm interested in the case where $A=B= M_n(\mathbb{C})$. So the theorem claims that given any two ...
1
vote
0answers
64 views

How to solve this nonlinear matrix equation

I would like to solve for $A$ is the following equation. $$ A^{-1} = B + \sum_{k=1}^d\sum_{l=1}^d A_{kl} C^{(kl)} $$ where $A$ is a $d\times d$ positive semi-definite matrix $B$ is a $d\times d$ ...
0
votes
2answers
48 views

linear map question

Prove that $f: \Bbb{R}^2 \to \Bbb{R}^2$ is a linear map if and only if there exists $a,b,c,d \in \Bbb{R}$ such that $f$ acts like multiplication by the matrix $$\left( \begin{array}{cc} a & b \\ ...
1
vote
1answer
78 views

Inverse a matrix $B+\lambda C$, where $\lambda$ is variable.

In my research I need to calculate $\operatorname{Trace}(A^{-1} C)$ where $A$ is given by two large, but sparse, matrices $B$ and $C$ by $A=B+\lambda C$. I need to do this inversion many times, so ...
0
votes
2answers
34 views

What is the term to make one matrix from two or more?

I am looking for the proper term for the operation of creating one block matrix from two or more for example $[AB]$ from $A$, $B$. And what is the correct notation to denote such a matrix. Do we use a ...
0
votes
1answer
82 views

Extremum of a multidimensional quadratic function

I have the following function: $$ g(h) = h'\Sigma\Sigma'h-h'm-r, $$ where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
7
votes
3answers
3k views

Solving non square matrix equations

Lets say we have: $\mathbf{A=BX}$ Where A and B are known matrices, X is unknown. In case B was square, a solution can be found by $\mathbf{B^{-1}A=X}$. But how do you attempt to solve for X when ...
4
votes
2answers
152 views

Prove that the set of all diagonal matrices is a subring of $\operatorname{Mat}_n(R)$ which is isomorphic to $R \times\dots\times R$ ($n$ factors)

Can someone tell me, is that diagonal matrices is a subring of $\operatorname{Mat}_n(R)$ which is (ring) isomorphic to $R \times · · · \times R$ (n factors) and why?.
1
vote
1answer
224 views

Questions about unipotent matrices

I'm reading Lang's Algebra. There is an example on page 19. Let $k$ be a field. Let $I$ be the unit $n\times n$ matrix, let $N$ be the additive group of matrices which are zero on and below the ...
0
votes
1answer
510 views

Simultaneous Equations That Should Be Inconsistent Has a Unique Solution

Find the values of $k$ for which the simultaneous equations do not have a unique solution for $x, y$ and $z$. Also show that when $k = -2$ the equations are inconsistent $$kx + 2y +z =0$$ $$3x + 0y ...
2
votes
1answer
1k views

inverse of diagonal plus sum of rank one matrices

Is there formula for the inverse of a matrix which is diagonal plus a sum of rank one matrices? $$S=\alpha I + \sum_1^N u_iu_i^T$$ $$S^{-1} = ?$$ Is there any decomposition or special trick that I ...
3
votes
1answer
91 views

Does $S^2=J_n^2(\lambda) $ imply $S=J_n(\lambda)$?

Assume $S \in M_{n\times n}(\Bbb{R})$ ,and $J_n(\lambda)$ denotes the Jordan block($\lambda \in \Bbb{R}^+$). If eigenvalues of $S$ are all positive real numbers. Does $$S^2=J_n^2(\lambda)$$ imply ...
2
votes
1answer
144 views

Variance Covariance Matrix, positive definiteness

Suppose we have a variance covariance matrix $\Sigma$. Under what conditions on the variance covariance matrix, $\Sigma$ is positive definite, that is $\forall w \neq 0, w^T \Sigma w>0$. In fact, ...
0
votes
2answers
263 views

Given a matrix factored into a product, how do you determine the determinant?

I'm preseneted with the question: Suppose that a 3x3 matrix A factors into the product of the two matrices below: \begin{matrix} 1 & 0 & 0 \\ I21 & 1 & 0 \\ I32 & I32 & ...
2
votes
0answers
114 views

What are the real world uses of Eigenbasis

The title pretty much says it all, I am wondering what the real world application (especially pertaining to electrical engineering) of an Eigenbasis is. I am also having some trouble understanding ...
5
votes
0answers
95 views

Complex Numbers vs. Matrix

I have a line starting at the origin, and i extend it to a point $(a,b)$ in the plane. This thing can be called a vector and be represented as $(a,b), [a\text{ }b]^T$ (column vector) or by ...
4
votes
1answer
374 views

Annihilators in matrix rings

Let $R$ be a finite commutative ring. For $n>1$ consider the full matrix ring $M_n(R)$. For a matrix $A\in M_n(R)$ is true that the cardinality of the left annihilator (in $M_n(R)$) of $A$ equals ...
0
votes
1answer
35 views

About invariant spaces

Let $A\in M_{n}$ and $W\subseteq\mathbb{C}^n$ be a subspace, such that $\textrm{dim}(W)\geq 1$. If $W$ is $A$-invariant, then $A$ has an eigenvector in $W$. I don't know how to prove it.
4
votes
1answer
101 views

Matrices that commute with two distinct matrices

There have been many questions in the vein of this one, but I can't find one that answers it specifically. Suppose $A,B\in M_n(\mathbb C)$ are two matrices such that, for any other $C\in M_n(\mathbb ...
2
votes
2answers
81 views

How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole

This seems pretty trivial but I'm not sure what to do. My coordinates are Cartesian, and I want to send point the $(x_1,...,x_{n-1},x_n)$ to point $(0,...,0,-1)$ so that all the other points are also ...
0
votes
1answer
34 views

Transform matrix to zero diagonals

Given a Hermitian positive semidefinite matrix $A$, is it possible to find a unitary matrix $U$ such that $UAU^H$ has zeros along the diagonal?
1
vote
3answers
71 views

Expressing a $3\times 3$ determinant as the product of four factors

I am attempting to express the determinant below as a product of four linear factors $$\begin{vmatrix} a & bc & b+c\\ b & ca & c+a\\ c & ab & a+b\\ \end{vmatrix} = ...
1
vote
3answers
160 views

Are there programs out there that try to derive linear recurrence given a string of numbers?

Wolfram isn't helping me much so I am curious if there are other programs out there. I don't know what degree it is, but I have a series of numbers and I'd like to determine the linear recurrence ...
1
vote
1answer
30 views

Iterative Power Regression

If I have a set of data points that would fit inside a power equation of the form y = a*x^b, what is the best ITERATIVE method to find the values of 'a' and 'b'. I thought I could compute the error ...